Pairing correlations within the micro-macroscopic approach for the level density

Abstract

Level density $\rho (E,N,Z)$ is calculated for the two-component close- and open-shell nuclei with a given energy E, and neutron N and proton Z numbers, taking into account pairing effects within the microscopic-macroscopic approach (MMA). These analytical calculations have been carried out by using semiclassical statistical mean-field approximations beyond the saddle-point method of the Fermi gas model in a low excitation-energies range. The level density $\rho$, obtained as function of the system entropy S, depends essentially on the condensation energy $E_{\textrm{cond}}$ through the excitation energy U in super-fluid nuclei. The simplest super-fluid approach, based on the BCS theory, accounts for a smooth temperature dependence of the pairing gap $\Delta$ due to particle number fluctuations. Taking into account the pairing effects in magic or semi-magic nuclei, excited below neutron resonances, one finds a notable pairing phase transition. Pairing correlations sometimes improve significantly the comparison with experimental data.

Appendix A: Semiclassical periodic orbit theory

For the sake of a simple notation, we shall presently restrict ourselves to the case of A nucleons (one kind only) in a given local (HF) potential $V(\textbf{r})$ as in Refs. [35, 48, 50,51,52]. The level density shell corrections can be presented analytically within the periodic orbit theory (POT) in terms of the sum over classical periodic orbits (PO) [48, 50, 51],

$$\begin{aligned}{} & {} {\displaystyle \delta g_{\textrm{scl}}(\varepsilon )= \sum ^{}_{\textrm{PO}}g^{}_{\textrm{PO}}(\varepsilon ),}\nonumber \\{} & {} {\displaystyle g^{}_{\textrm{PO}}(\varepsilon )\!=\!{\mathcal {A}}_\textrm{PO}(\varepsilon ) \,\cos \left[ \frac{1}{\hbar }{\mathbb {S}}_\textrm{PO}(\varepsilon )- \frac{\pi }{2} \mu ^{}_{\textrm{PO}} -\phi ^{}_0\right] \!.} \end{aligned}$$

(A1)

Here ${\mathbb {S}}_{\textrm{PO}}(\varepsilon )$ is the classical action along the PO in the nucleon potential well of the same radius, $R=r_0A^{1/3}$ ($r^{}_0\approx 1.14$ fm), $\mu ^{}_{\textrm{PO}}$ is the so called Maslov index, determined by the catastrophe points (turning and caustic points) along the PO, and $\phi ^{}_0$ is an additional shift of the phase coming from the dimension of the problem and degeneracy of the POs. The amplitude ${\mathcal {A}}_\textrm{PO}(\varepsilon )$, and the action ${\mathbb {S}}_\textrm{PO}(\varepsilon )$, are smooth functions of the energy $\varepsilon$. In addition, the amplitude, ${\mathcal {A}}_{\textrm{PO}}(\varepsilon )$, depends on the PO stability factors. The Gaussian local averaging of the level density shell correction, $\delta g_\textrm{scl}(\varepsilon )$, over the single-particle (s.p.) energy spectrum $\varepsilon ^{}_i$ near the Fermi surface $\varepsilon ^{}_F$, with a width parameter $\gamma$, smaller than a distance between major shells, ${\mathcal {D}}_{\textrm{sh}}$, can be done analytically [48, 50, 51],

$$\begin{aligned} \delta g^{}_{\gamma \mathrm scl}(\varepsilon ) \cong \sum ^{}_\textrm{PO}g^{}_{\textrm{PO}}(\varepsilon )~ \exp \left[ -\left( \frac{\gamma t^{}_{\textrm{PO}}}{2\hbar }\right) ^2\right] ~, \end{aligned}$$

(A2)

where $t^{}_{\textrm{PO}}= \partial {\mathbb {S}}_{\textrm{PO}}/\partial \varepsilon$ is the period of particle motion along the PO in the potential well.

The smooth ground-state energy of the nucleus is approximated by ${\tilde{E}}\approx E_{\mathrm{\texttt{ETF}}}=\int _0^{\lambda } \hbox {d}\varepsilon ~\varepsilon ~ {\tilde{g}}(\varepsilon )$ , where ${\tilde{g}}(\varepsilon )$ is a smooth level density equal approximately to the ETF level density, ${\tilde{g}}\approx g_\mathrm{\texttt{ETF}}$, ($\lambda \approx {\tilde{\lambda }}$, and ${\tilde{\lambda }}$ is the smooth chemical potential in the SCM). The chemical potential $\lambda$ (or $\tilde{\lambda }$) is the solution of the corresponding conservation of particle number equation:

$$\begin{aligned} A = \int \limits _0^{\lambda }\text{ d } \varepsilon ~g(\varepsilon ) ~. \qquad \end{aligned}$$

(A3)

The POT shell component of the free energy, $\delta F_{\textrm{scl}}$, is related in the non-thermal and non-rotational limit to the shell correction energy of a cold nucleus, $\delta E_{\textrm{scl}}$, see Refs. [48, 50, 51]. Within the POT, $\delta E_{\textrm{scl}}$ is determined, in turn, by the oscillating level density $\delta g_\textrm{scl}(\varepsilon )$, Eq. (A1),

$$\begin{aligned} \delta E_{\textrm{scl}} \approx \sum _{\textrm{PO}}\frac{\hbar ^2}{t_\textrm{PO}^2}\delta g^{}_{\textrm{PO}}(\lambda )~. \end{aligned}$$

(A4)

The chemical potential $\lambda$ can be approximated by the Fermi energy $\varepsilon ^{}_F$, up to a small excitation energy, and isotopic asymmetry corrections ($T\ll \lambda$ for the saddle point value $T=1/\beta ^*$, if exists). It is determined by the particle-number conservation conditions, Eq.  (A3), where $g(\varepsilon )\cong g_{\textrm{scl}}= g^{}_{\mathrm{\texttt{ETF}}} +\delta g_{\textrm{scl}}$ is the total POT level density. One now needs to solve equation (A3) to determine the chemical potential $\lambda$ as function of the particle number A since $\lambda$ is needed in Eq. (A4) to obtain the semiclassical energy shell correction $\delta E_{\textrm{scl}}$.

For a major shell structure near the Fermi energy surface $\varepsilon \approx \lambda$, the POT energy shell correction, $\delta E_{\textrm{scl}}$, is approximately proportional to the level density shell correction, $\delta g_{\textrm{scl}}(\varepsilon )$ [Eq. (A1)], at $\varepsilon =\lambda$, Eq. (A4),

$$\begin{aligned} \delta E \approx \delta E_{\textrm{scl}}\approx \left( \frac{{\mathcal {D}}_{\textrm{sh}}}{2 \pi }\right) ^2~ \delta g_\textrm{scl}(\lambda )~, \end{aligned}$$

(A5)

where ${\mathcal {D}}_{\textrm{sh}} \approx \lambda /A^{1/3}$ is the mean distance between major nuclear shells. Indeed, the rapid convergence of the PO sum in Eq. (A4) is guaranteed by the factor in front of the density component $g^{}_{\textrm{PO}}$, Eq. (A1), a factor which is inversely proportional to the period time $t_{\textrm{PO}}(\lambda )$ squared along the PO. Therefore, only POs with short periods which occupy a significant phase-space volume near the Fermi surface will contribute. These orbits are responsible for the major shell structure, that is related to a Gaussian averaging width, $\gamma \approx \gamma _{\textrm{sh}}$, which is much larger than the distance between neighboring s.p. states but much smaller than the distance ${\mathcal {D}}_{\textrm{sh}}$ between major shells near the Fermi surface. Eq. (A2) for the averaged s.p. level density was derived under these conditions for $\gamma$. According to the POT [48, 50, 51], the distance between major shells, ${\mathcal {D}}_{\textrm{sh}}$, is determined by a mean period of the shortest and most degenerate POs, $\langle t^{}_\textrm{PO}\rangle$ [50]:

$$\begin{aligned} {\mathcal {D}}_{\textrm{sh}} \cong \frac{2\pi \hbar }{\langle t_\textrm{PO}\rangle } \approx \frac{\lambda }{A^{1/3}}~. \end{aligned}$$

(A6)

Taking the factor in front of $g^{}_{\textrm{PO}}$, in Eq. (A4) off the sum over the POs for the energy shell correction $\delta E_{\textrm{scl}}$, one arrives at its semiclassical expression (A5) [48, 49, 51]. Differentiating Eq. (A5) with (A1) with respect to $\lambda$ and keeping only the dominating terms coming from differentiation of the sine of the action phase argument, ${\mathbb {S}}_{\textrm{PO}}/\hbar \sim A^{1/3}$, one finds the useful relationships:

$$\begin{aligned}{} & {} {\displaystyle \frac{\partial ^2\delta E_\textrm{PO}}{\partial \lambda ^2} \approx -\delta g^{}_{\textrm{PO}}(\lambda )~},\nonumber \\{} & {} {\displaystyle \frac{\partial ^2 g_\textrm{scl}}{\partial \lambda ^2}\approx \sum ^{}_\textrm{PO}\frac{\partial ^2\delta g^{}_{\textrm{PO}}}{\partial \lambda ^2} \approx -\left( \frac{2\pi }{D_{\textrm{sh}}}\right) ^2 \delta g^{}_\textrm{PO}(\lambda )~}. \end{aligned}$$

(A7)

Notice that taking into account Eq. (A3) for the chemical potential $\lambda$, one has another useful relationship for the second derivative of background thermodynamic potential, $\Omega _0=\int _0^\lambda \hbox {d}\varepsilon (\varepsilon -\lambda ) g_\textrm{scl}(\varepsilon )$:

$$\begin{aligned} \frac{\partial ^2\Omega _{0}}{\partial \lambda ^2} \approx - g^{}_\textrm{scl}(\lambda )~. \end{aligned}$$

(A8)

The level density parameter a [see Eq. (10) for the entropy S] can be related to the averaged POT level density ETF and shell correction components by

$$\begin{aligned} a \approx a^{}_{\textrm{ETF}}+\delta a_{\textrm{scl}}= \frac{\pi ^2}{6} (g^{}_{\textrm{ETF}}+\delta g_{\textrm{scl}})~, \end{aligned}$$

(A9)

For shell corrections, one has the following relation:

$$\begin{aligned} \delta a_{\textrm{scl}}=\frac{\pi ^2}{6}\delta g_{\textrm{scl}}(\lambda ). \end{aligned}$$

(A10)

Using Eqs. (A9), (A10), and Eq. (A7), one obtains Eq. (9).